Examinando por Autor "Lillo, R.E."

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    Directional multivariate extremes in environmental phenomena
    (John Wiley and Sons Ltd, 2017-03-01) Torres, R.; De michele, C.; Laniado, H.; Lillo, R.E.; Universidad EAFIT. Escuela de Ciencias; Modelado Matemático
    Several environmental phenomena can be described by different correlated variables that must be considered jointly in order to be more representative of the nature of these phenomena. For such events, identification of extremes is inappropriate if it is based on marginal analysis. Extremes have usually been linked to the notion of quantile, which is an important tool to analyze risk in the univariate setting. We propose to identify multivariate extremes and analyze environmental phenomena in terms of the directional multivariate quantile, which allows us to analyze the data considering all the variables implied in the phenomena, as well as look at the data in interesting directions that can better describe an environmental catastrophe. Because there are many references in the literature that propose extremes detection based on copula models, we also generalize the copula method by introducing the directional approach. Advantages and disadvantages of the nonparametric proposal that we introduce and the copula methods are provided in the paper. We show with simulated and real data sets how by considering the first principal component direction we can improve the visualization of extremes. Finally, two cases of study are analyzed: a synthetic case of flood risk at a dam (a three-variable case) and a real case study of sea storms (a five-variable case). Copyright © 2017 John Wiley & Sons, Ltd.
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    A directional multivariate value at risk
    (Elsevier, 2015-11-01) Torres, R.; Lillo, R.E.; Laniado, H.; Universidad EAFIT. Escuela de Ciencias; Modelado Matemático
    In economics, insurance and finance, value at risk (VaR) is a widely used measure of the risk of loss on a specific portfolio of financial assets. For a given portfolio, time horizon, and probability alpha, the 100 alpha% VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is alpha. That is to say, it is a quantile of the distribution of the losses, which has both good analytic properties and easy interpretation as a risk measure. However, its extension to the multivariate framework is not unique because a unique definition of multivariate quantile does not exist. In the current literature, the multivariate quantiles are related to a specific partial order considered in R-n, or to a property of the univariate quantile that is desirable to be extended to R-n. In this work, we introduce a multivariate value at risk as a vector-valued directional risk measure, based on a directional multivariate quantile, which has recently been introduced in the literature. The directional approach allows the manager to consider external information or risk preferences in her/his analysis. We derive some properties of the risk measure and we compare the univariate VaR over the marginals with the components of the directional multivariate VaR. We also analyze the relationship between some families of copulas, for which it is possible to obtain closed forms of the multivariate VaR that we propose. Finally, comparisons with other alternative multivariate VaR given in the literature, are provided in terms of robustness. (C) 2015 Elsevier B.V. All rights reserved.